A \(q\) -analog of isomonodromic deformation of linear differential equations can be considered. It corresponds to study connection preserved deformation of linear \(q\) -difference equations, that is, \(q\) -difference Lax pairs with rational coefficients of the independent variable. In [1]}, the \(q\) -Painlevé VI (\(q\) -\(\mathrm {P_{VI}}\) ) was obtained from the compatibility condition of a Lax pair of rank 2. In [2]}, fundamental solutions of the Lax pair were constructed in terms of the \(q\) -conformal blocks given as expectation values of the intertwining operators of the Ding–Iohara–Miki algebra [3]}, and expressed in terms of the \(q\) -Nekrasov functions. As a result, a general solution to the \(q\) -Painlevé VI is expressed explicitly, and moreover, it was conjectured that tau functions of \(q\) -\(\mathrm {P_{VI}}\) defined as Fourier transforms of the \(q\) -conformal blocks satisfy bilinear equations. Later, by taking degeneration of \(q\) -\(\mathrm {P_{VI}}\) , explicit series representations for the tau functions of the \(q\) -Painlevé V, \(\mathrm {III_1}\) , \(\mathrm {III_2}\) , and \(\mathrm {III_3}\)  equations were obtained [4]}. The series representations of the tau function for \(q\) -\(\mathrm {P_{{III}_3}}\)  expressed by the pure 5d Nekrasov function had been proposed in [5]} for surface type \(A_7^{(1)\prime }\)  and [6]} for surface type \(A_7^{(1)}\) . The tau functions obtained in [4]} are equivalent to those proposed in [6]}, [5]}. A Fredholm determinant representation for the tau functions of \(q\) -\(\mathrm {P_{III_3}}\) was presented in [10]}.
